author | haftmann |
Mon, 29 Sep 2008 12:31:58 +0200 | |
changeset 28401 | d5f39173444c |
parent 27556 | 292098f2efdf |
child 30042 | 31039ee583fa |
permissions | -rw-r--r-- |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
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1 |
(* Title: HOL/Quadratic_Reciprocity/Gauss.thy |
14981 | 2 |
ID: $Id$ |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
3 |
Authors: Jeremy Avigad, David Gray, and Adam Kramer |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
4 |
*) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
5 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
6 |
header {*Integers: Divisibility and Congruences*} |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
7 |
|
27556 | 8 |
theory Int2 |
9 |
imports Finite2 WilsonRuss |
|
10 |
begin |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
11 |
|
19670 | 12 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20217
diff
changeset
|
13 |
MultInv :: "int => int => int" where |
19670 | 14 |
"MultInv p x = x ^ nat (p - 2)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
15 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
16 |
|
19670 | 17 |
subsection {* Useful lemmas about dvd and powers *} |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
18 |
|
18369 | 19 |
lemma zpower_zdvd_prop1: |
20 |
"0 < n \<Longrightarrow> p dvd y \<Longrightarrow> p dvd ((y::int) ^ n)" |
|
21 |
by (induct n) (auto simp add: zdvd_zmult zdvd_zmult2 [of p y]) |
|
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
22 |
|
18369 | 23 |
lemma zdvd_bounds: "n dvd m ==> m \<le> (0::int) | n \<le> m" |
24 |
proof - |
|
25 |
assume "n dvd m" |
|
26 |
then have "~(0 < m & m < n)" |
|
27 |
using zdvd_not_zless [of m n] by auto |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
28 |
then show ?thesis by auto |
18369 | 29 |
qed |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
30 |
|
19670 | 31 |
lemma zprime_zdvd_zmult_better: "[| zprime p; p dvd (m * n) |] ==> |
18369 | 32 |
(p dvd m) | (p dvd n)" |
33 |
apply (cases "0 \<le> m") |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
34 |
apply (simp add: zprime_zdvd_zmult) |
18369 | 35 |
apply (insert zprime_zdvd_zmult [of "-m" p n]) |
36 |
apply auto |
|
37 |
done |
|
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
38 |
|
18369 | 39 |
lemma zpower_zdvd_prop2: |
40 |
"zprime p \<Longrightarrow> p dvd ((y::int) ^ n) \<Longrightarrow> 0 < n \<Longrightarrow> p dvd y" |
|
41 |
apply (induct n) |
|
42 |
apply simp |
|
43 |
apply (frule zprime_zdvd_zmult_better) |
|
44 |
apply simp |
|
45 |
apply force |
|
46 |
done |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
47 |
|
18369 | 48 |
lemma div_prop1: "[| 0 < z; (x::int) < y * z |] ==> x div z < y" |
49 |
proof - |
|
23315 | 50 |
assume "0 < z" then have modth: "x mod z \<ge> 0" by simp |
51 |
have "(x div z) * z \<le> (x div z) * z" by simp |
|
52 |
then have "(x div z) * z \<le> (x div z) * z + x mod z" using modth by arith |
|
53 |
also have "\<dots> = x" |
|
18369 | 54 |
by (auto simp add: zmod_zdiv_equality [symmetric] zmult_ac) |
55 |
also assume "x < y * z" |
|
56 |
finally show ?thesis |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
13871
diff
changeset
|
57 |
by (auto simp add: prems mult_less_cancel_right, insert prems, arith) |
18369 | 58 |
qed |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
59 |
|
18369 | 60 |
lemma div_prop2: "[| 0 < z; (x::int) < (y * z) + z |] ==> x div z \<le> y" |
61 |
proof - |
|
62 |
assume "0 < z" and "x < (y * z) + z" |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
63 |
then have "x < (y + 1) * z" by (auto simp add: int_distrib) |
18369 | 64 |
then have "x div z < y + 1" |
65 |
apply - |
|
66 |
apply (rule_tac y = "y + 1" in div_prop1) |
|
67 |
apply (auto simp add: prems) |
|
68 |
done |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
69 |
then show ?thesis by auto |
18369 | 70 |
qed |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
71 |
|
18369 | 72 |
lemma zdiv_leq_prop: "[| 0 < y |] ==> y * (x div y) \<le> (x::int)" |
73 |
proof- |
|
74 |
assume "0 < y" |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
75 |
from zmod_zdiv_equality have "x = y * (x div y) + x mod y" by auto |
18369 | 76 |
moreover have "0 \<le> x mod y" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
77 |
by (auto simp add: prems pos_mod_sign) |
18369 | 78 |
ultimately show ?thesis |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
79 |
by arith |
18369 | 80 |
qed |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
81 |
|
19670 | 82 |
|
83 |
subsection {* Useful properties of congruences *} |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
84 |
|
18369 | 85 |
lemma zcong_eq_zdvd_prop: "[x = 0](mod p) = (p dvd x)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
86 |
by (auto simp add: zcong_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
87 |
|
18369 | 88 |
lemma zcong_id: "[m = 0] (mod m)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
89 |
by (auto simp add: zcong_def zdvd_0_right) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
90 |
|
18369 | 91 |
lemma zcong_shift: "[a = b] (mod m) ==> [a + c = b + c] (mod m)" |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
92 |
by (auto simp add: zcong_refl zcong_zadd) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
93 |
|
18369 | 94 |
lemma zcong_zpower: "[x = y](mod m) ==> [x^z = y^z](mod m)" |
95 |
by (induct z) (auto simp add: zcong_zmult) |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
96 |
|
19670 | 97 |
lemma zcong_eq_trans: "[| [a = b](mod m); b = c; [c = d](mod m) |] ==> |
18369 | 98 |
[a = d](mod m)" |
99 |
apply (erule zcong_trans) |
|
100 |
apply simp |
|
101 |
done |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
102 |
|
18369 | 103 |
lemma aux1: "a - b = (c::int) ==> a = c + b" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
104 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
105 |
|
19670 | 106 |
lemma zcong_zmult_prop1: "[a = b](mod m) ==> ([c = a * d](mod m) = |
18369 | 107 |
[c = b * d] (mod m))" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
108 |
apply (auto simp add: zcong_def dvd_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
109 |
apply (rule_tac x = "ka + k * d" in exI) |
18369 | 110 |
apply (drule aux1)+ |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
111 |
apply (auto simp add: int_distrib) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
112 |
apply (rule_tac x = "ka - k * d" in exI) |
18369 | 113 |
apply (drule aux1)+ |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
114 |
apply (auto simp add: int_distrib) |
18369 | 115 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
116 |
|
19670 | 117 |
lemma zcong_zmult_prop2: "[a = b](mod m) ==> |
18369 | 118 |
([c = d * a](mod m) = [c = d * b] (mod m))" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
119 |
by (auto simp add: zmult_ac zcong_zmult_prop1) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
120 |
|
19670 | 121 |
lemma zcong_zmult_prop3: "[| zprime p; ~[x = 0] (mod p); |
18369 | 122 |
~[y = 0] (mod p) |] ==> ~[x * y = 0] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
123 |
apply (auto simp add: zcong_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
124 |
apply (drule zprime_zdvd_zmult_better, auto) |
18369 | 125 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
126 |
|
19670 | 127 |
lemma zcong_less_eq: "[| 0 < x; 0 < y; 0 < m; [x = y] (mod m); |
18369 | 128 |
x < m; y < m |] ==> x = y" |
25675 | 129 |
by (metis zcong_not zcong_sym zless_linear) |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
130 |
|
19670 | 131 |
lemma zcong_neg_1_impl_ne_1: "[| 2 < p; [x = -1] (mod p) |] ==> |
18369 | 132 |
~([x = 1] (mod p))" |
133 |
proof |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
134 |
assume "2 < p" and "[x = 1] (mod p)" and "[x = -1] (mod p)" |
18369 | 135 |
then have "[1 = -1] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
136 |
apply (auto simp add: zcong_sym) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
137 |
apply (drule zcong_trans, auto) |
18369 | 138 |
done |
139 |
then have "[1 + 1 = -1 + 1] (mod p)" |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
140 |
by (simp only: zcong_shift) |
18369 | 141 |
then have "[2 = 0] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
142 |
by auto |
18369 | 143 |
then have "p dvd 2" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
144 |
by (auto simp add: dvd_def zcong_def) |
18369 | 145 |
with prems show False |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
146 |
by (auto simp add: zdvd_not_zless) |
18369 | 147 |
qed |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
148 |
|
18369 | 149 |
lemma zcong_zero_equiv_div: "[a = 0] (mod m) = (m dvd a)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
150 |
by (auto simp add: zcong_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
151 |
|
19670 | 152 |
lemma zcong_zprime_prod_zero: "[| zprime p; 0 < a |] ==> |
153 |
[a * b = 0] (mod p) ==> [a = 0] (mod p) | [b = 0] (mod p)" |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
154 |
by (auto simp add: zcong_zero_equiv_div zprime_zdvd_zmult) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
155 |
|
16663 | 156 |
lemma zcong_zprime_prod_zero_contra: "[| zprime p; 0 < a |] ==> |
18369 | 157 |
~[a = 0](mod p) & ~[b = 0](mod p) ==> ~[a * b = 0] (mod p)" |
19670 | 158 |
apply auto |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
159 |
apply (frule_tac a = a and b = b and p = p in zcong_zprime_prod_zero) |
18369 | 160 |
apply auto |
161 |
done |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
162 |
|
19670 | 163 |
lemma zcong_not_zero: "[| 0 < x; x < m |] ==> ~[x = 0] (mod m)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
164 |
by (auto simp add: zcong_zero_equiv_div zdvd_not_zless) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
165 |
|
18369 | 166 |
lemma zcong_zero: "[| 0 \<le> x; x < m; [x = 0](mod m) |] ==> x = 0" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
167 |
apply (drule order_le_imp_less_or_eq, auto) |
18369 | 168 |
apply (frule_tac m = m in zcong_not_zero) |
169 |
apply auto |
|
170 |
done |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
171 |
|
27556 | 172 |
lemma all_relprime_prod_relprime: "[| finite A; \<forall>x \<in> A. zgcd x y = 1 |] |
173 |
==> zgcd (setprod id A) y = 1" |
|
22274 | 174 |
by (induct set: finite) (auto simp add: zgcd_zgcd_zmult) |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
175 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
176 |
|
19670 | 177 |
subsection {* Some properties of MultInv *} |
178 |
||
179 |
lemma MultInv_prop1: "[| 2 < p; [x = y] (mod p) |] ==> |
|
18369 | 180 |
[(MultInv p x) = (MultInv p y)] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
181 |
by (auto simp add: MultInv_def zcong_zpower) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
182 |
|
19670 | 183 |
lemma MultInv_prop2: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 184 |
[(x * (MultInv p x)) = 1] (mod p)" |
185 |
proof (simp add: MultInv_def zcong_eq_zdvd_prop) |
|
186 |
assume "2 < p" and "zprime p" and "~ p dvd x" |
|
187 |
have "x * x ^ nat (p - 2) = x ^ (nat (p - 2) + 1)" |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
188 |
by auto |
18369 | 189 |
also from prems have "nat (p - 2) + 1 = nat (p - 2 + 1)" |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19670
diff
changeset
|
190 |
by (simp only: nat_add_distrib) |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
191 |
also have "p - 2 + 1 = p - 1" by arith |
18369 | 192 |
finally have "[x * x ^ nat (p - 2) = x ^ nat (p - 1)] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
193 |
by (rule ssubst, auto) |
18369 | 194 |
also from prems have "[x ^ nat (p - 1) = 1] (mod p)" |
19670 | 195 |
by (auto simp add: Little_Fermat) |
18369 | 196 |
finally (zcong_trans) show "[x * x ^ nat (p - 2) = 1] (mod p)" . |
197 |
qed |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
198 |
|
19670 | 199 |
lemma MultInv_prop2a: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 200 |
[(MultInv p x) * x = 1] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
201 |
by (auto simp add: MultInv_prop2 zmult_ac) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
202 |
|
18369 | 203 |
lemma aux_1: "2 < p ==> ((nat p) - 2) = (nat (p - 2))" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
diff
changeset
|
204 |
by (simp add: nat_diff_distrib) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
diff
changeset
|
205 |
|
18369 | 206 |
lemma aux_2: "2 < p ==> 0 < nat (p - 2)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
207 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
208 |
|
19670 | 209 |
lemma MultInv_prop3: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 210 |
~([MultInv p x = 0](mod p))" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
211 |
apply (auto simp add: MultInv_def zcong_eq_zdvd_prop aux_1) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
212 |
apply (drule aux_2) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
213 |
apply (drule zpower_zdvd_prop2, auto) |
18369 | 214 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
215 |
|
19670 | 216 |
lemma aux__1: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==> |
217 |
[(MultInv p (MultInv p x)) = (x * (MultInv p x) * |
|
18369 | 218 |
(MultInv p (MultInv p x)))] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
219 |
apply (drule MultInv_prop2, auto) |
18369 | 220 |
apply (drule_tac k = "MultInv p (MultInv p x)" in zcong_scalar, auto) |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
221 |
apply (auto simp add: zcong_sym) |
18369 | 222 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
223 |
|
16663 | 224 |
lemma aux__2: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==> |
18369 | 225 |
[(x * (MultInv p x) * (MultInv p (MultInv p x))) = x] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
226 |
apply (frule MultInv_prop3, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
227 |
apply (insert MultInv_prop2 [of p "MultInv p x"], auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
228 |
apply (drule MultInv_prop2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
229 |
apply (drule_tac k = x in zcong_scalar2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
230 |
apply (auto simp add: zmult_ac) |
18369 | 231 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
232 |
|
19670 | 233 |
lemma MultInv_prop4: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 234 |
[(MultInv p (MultInv p x)) = x] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
235 |
apply (frule aux__1, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
236 |
apply (drule aux__2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
237 |
apply (drule zcong_trans, auto) |
18369 | 238 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
239 |
|
19670 | 240 |
lemma MultInv_prop5: "[| 2 < p; zprime p; ~([x = 0](mod p)); |
241 |
~([y = 0](mod p)); [(MultInv p x) = (MultInv p y)] (mod p) |] ==> |
|
18369 | 242 |
[x = y] (mod p)" |
19670 | 243 |
apply (drule_tac a = "MultInv p x" and b = "MultInv p y" and |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
244 |
m = p and k = x in zcong_scalar) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
245 |
apply (insert MultInv_prop2 [of p x], simp) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
246 |
apply (auto simp only: zcong_sym [of "MultInv p x * x"]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
247 |
apply (auto simp add: zmult_ac) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
248 |
apply (drule zcong_trans, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
249 |
apply (drule_tac a = "x * MultInv p y" and k = y in zcong_scalar, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
250 |
apply (insert MultInv_prop2a [of p y], auto simp add: zmult_ac) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
251 |
apply (insert zcong_zmult_prop2 [of "y * MultInv p y" 1 p y x]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
252 |
apply (auto simp add: zcong_sym) |
18369 | 253 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
254 |
|
19670 | 255 |
lemma MultInv_zcong_prop1: "[| 2 < p; [j = k] (mod p) |] ==> |
18369 | 256 |
[a * MultInv p j = a * MultInv p k] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
257 |
by (drule MultInv_prop1, auto simp add: zcong_scalar2) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
258 |
|
19670 | 259 |
lemma aux___1: "[j = a * MultInv p k] (mod p) ==> |
18369 | 260 |
[j * k = a * MultInv p k * k] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
261 |
by (auto simp add: zcong_scalar) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
262 |
|
19670 | 263 |
lemma aux___2: "[|2 < p; zprime p; ~([k = 0](mod p)); |
18369 | 264 |
[j * k = a * MultInv p k * k] (mod p) |] ==> [j * k = a] (mod p)" |
19670 | 265 |
apply (insert MultInv_prop2a [of p k] zcong_zmult_prop2 |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
266 |
[of "MultInv p k * k" 1 p "j * k" a]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
267 |
apply (auto simp add: zmult_ac) |
18369 | 268 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
269 |
|
19670 | 270 |
lemma aux___3: "[j * k = a] (mod p) ==> [(MultInv p j) * j * k = |
18369 | 271 |
(MultInv p j) * a] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
272 |
by (auto simp add: zmult_assoc zcong_scalar2) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
273 |
|
19670 | 274 |
lemma aux___4: "[|2 < p; zprime p; ~([j = 0](mod p)); |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
275 |
[(MultInv p j) * j * k = (MultInv p j) * a] (mod p) |] |
18369 | 276 |
==> [k = a * (MultInv p j)] (mod p)" |
19670 | 277 |
apply (insert MultInv_prop2a [of p j] zcong_zmult_prop1 |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
278 |
[of "MultInv p j * j" 1 p "MultInv p j * a" k]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
279 |
apply (auto simp add: zmult_ac zcong_sym) |
18369 | 280 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
281 |
|
19670 | 282 |
lemma MultInv_zcong_prop2: "[| 2 < p; zprime p; ~([k = 0](mod p)); |
283 |
~([j = 0](mod p)); [j = a * MultInv p k] (mod p) |] ==> |
|
18369 | 284 |
[k = a * MultInv p j] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
285 |
apply (drule aux___1) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
286 |
apply (frule aux___2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
287 |
by (drule aux___3, drule aux___4, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
288 |
|
19670 | 289 |
lemma MultInv_zcong_prop3: "[| 2 < p; zprime p; ~([a = 0](mod p)); |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
290 |
~([k = 0](mod p)); ~([j = 0](mod p)); |
19670 | 291 |
[a * MultInv p j = a * MultInv p k] (mod p) |] ==> |
18369 | 292 |
[j = k] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
293 |
apply (auto simp add: zcong_eq_zdvd_prop [of a p]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
294 |
apply (frule zprime_imp_zrelprime, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
295 |
apply (insert zcong_cancel2 [of p a "MultInv p j" "MultInv p k"], auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
296 |
apply (drule MultInv_prop5, auto) |
18369 | 297 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
298 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
299 |
end |